where \(N\)  is a normalization constant. \(P_l^{(\alpha ,\beta )}(z)\)  is the Jacobi polynomial and \(\alpha = 2m_{\bar{q}}/\sigma \) , \(\beta = 2m_{q}/\sigma \) .
The obtained solutions are closely related to the 't Hooft model [1]}, [2]} as well as the collinear QCD in 1+1D [3]}. Immediately, the mass squared of the ground state (\(n=m=l=0,J=0\) ), i.e. the pion, is \(M_\pi ^2 = 2\sigma m_{u,d} + 4m_{u,d}^2\) , satisfying the GMOR relation with \(\sigma = |\langle \bar{q} q\rangle |/f^2_\pi \) , up to \(\mathrm {O}(m_q)\)  [4]}. The pion wave function now becomes,
\(\psi _\pi (x, \vec{k}_\perp ) = ({4\pi N}/{\kappa }) x^\frac{\mu }{2}(1-x)^\frac{\mu }{2}\exp \big [-{\vec{k}^2_\perp }/({2\kappa ^2x(1-x)})\big ]\) ,
where \(\mu = 2m_{u,d}/\sigma \) .
Fig. REF  compares the normalized ground-state longitudinal wave functions obtained from IMA and from this work in three light meson sectors, \(q\bar{q}\) , \(s\bar{s}\)  and \(s\bar{q}\) , where \(q=u,d\) . Visually, the longitudinal wave functions from these two approaches are nearly identical for the \(q\bar{q}\)  sector, but differ significantly for the \(s\bar{s}\)  and \(s\bar{q}\)  sectors. Note that the quark masses from these two approaches are remarkably different (see Table REF  and the relevant discussions in the main text). Major differences occur at the endpoints \(x\rightarrow 0,1\) , our wave functions are power-law like whereas the IMA wave functions are more suppressed. As a result, observables singular at the endpoints, e.g. the light-front kinetic energy and the form factors, acquire very different expectation values from these two sets of wave functions. The endpoint behavior also has a dramatic impact on hadronic observables in high energy collisions as hard kernels \(T_H\)  are sensitive to the endpoint singularities [5]}.
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